\[ \int \frac {x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {5 x}{128 a \,b^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {x^{5}}{8 b \left (b \,x^{2}+a \right )^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {5 x^{3}}{48 b^{2} \left (b \,x^{2}+a \right )^{2} \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {5 x}{64 b^{3} \left (b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {5 \left (b \,x^{2}+a \right ) \arctan \left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{128 a^{\frac {3}{2}} b^{\frac {7}{2}} \sqrt {\left (b \,x^{2}+a \right )^{2}}} \]
command
integrate(x^6/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {5 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {15 \, b^{3} x^{7} - 73 \, a b^{2} x^{5} - 55 \, a^{2} b x^{3} - 15 \, a^{3} x}{384 \, {\left (b x^{2} + a\right )}^{4} a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________